Hi @Roger Shrimpton ,
What type of account are you using? Is it an outlook.com or exchange or o365 account?
When your mailbox is nearing its maximum capacity, we will begin sending you emails and in-app messages telling you your storage is almost full. If it does exceed capacity, you won't be able to send or receive any messages. Also, people who send you email will receive an error message that tells you your mailbox is full.
To see how much mailbox storage, you've used and quickly clear up space: At the top of the page, select Settings > View all Outlook settings > General > Storage.
Refer to this article to check your mailbox storage and clean up mailboxes: Mailbox storage limits in Outlook on the web
When you no longer want to see messages from someone, you can block a sender via this link: Block a mail sender
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